Mathematicians in Bologna, 1861-1960

Cover......Page 1
Mathematicians in Bologna 1861–1960......Page 4
A Short Overview on Mathematicians in Bologna in the First Century after the Establishment of Italy......Page 6
References......Page 9
Contents......Page 10
On Cimmino Integrals as Residues of Zeta Functions......Page 12
2 Non-Euclidean Geometry Before Beltrami......Page 15
3 The Models of Beltrami......Page 18
3.1 The ``Projective'' Model......Page 19
3.1.1 Anisotropy......Page 24
3.2 The ``Conformal'' Models......Page 25
3.2.1 Imbedding the Pseudosphere in Euclidean Space......Page 28
3.2.2 Anachronism (for the Complex Reader)......Page 30
3.3 What was Beltrami's Interpretation of His Own Work?......Page 32
4 From the Boundary to the Interior: An Example from Signal Processing......Page 35
References......Page 40
Giuseppe Vitali: Real and Complex Analysisand Differential Geometry......Page 42
1 Formation and Career......Page 43
2 First Research Works in Complex Analysis......Page 46
3 Research on Real Analysis......Page 47
4 The Years of Secondary School Teaching......Page 54
5 Between the Two Wars: Differential Geometry......Page 55
6 Conclusions......Page 60
References......Page 61
1 Introduction......Page 67
2 Attending Weierstrass's Lectures......Page 68
3 Expansions in Series......Page 70
4 Functional Operations......Page 77
References......Page 80
1 The Beginnings......Page 89
2 The Academic Environment......Page 90
3 Teaching and Institutional Responsabilities......Page 98
4 Cremona's Studies and Research......Page 100
References......Page 113
1 Introduction......Page 115
2 The Route to Bologna......Page 116
3 First Lectures in Bologna......Page 119
4 The Lectures on Higher Geometry......Page 123
4.1 Segre's Influence......Page 125
4.2 Philosophical Reflections on Foundations of Geometry......Page 126
4.3 A Closer Look at the Contents of the Lecture Notes......Page 130
4.4 From the Lectures to the Enzyklopädie......Page 135
4.5 Towards the Questions About Elementary Mathematics......Page 137
5.1 The Contents of the Course......Page 138
5.2 A Few Comments......Page 140
6 Conclusions......Page 144
References......Page 146
1 Introduction......Page 153
2 The End of Our Story......Page 154
3 First Years in Rome......Page 155
4 First Digression: Corrado Segre......Page 157
5.1 Assistant in Rome and Pavia......Page 160
5.2 Professor in Milan......Page 161
6.1 Reaching Bologna......Page 163
6.2 Visits Abroad......Page 164
6.3 Becoming an Ordinary Professor......Page 165
7.1 Fubini's Idea: Differential Forms......Page 168
7.1.1 Fubini's First Two Fundamental Forms and the First Rigidity Theorem......Page 171
7.1.2 Fubini's Third Fundamental Form and the Main Rigidity Theorem......Page 172
8 Some Glimpses to Bompiani's Work in the Bologna Period......Page 173
9 Back to Rome......Page 179
10.1 Bompiani in the Following Years......Page 180
10.2 Bompiani and the Fascist Regime......Page 181
11 Conclusion......Page 182
References......Page 183
Dario Graffi in a Complex Historical Period......Page 188
1 Vector Calculus in Mechanics Treatises in Italy in the Early Twentieth Century......Page 206
2 The Vector Calculus of Burgattiand of Burali–Forti–Marcolongo......Page 207
3 Burgatti's Treatise on Mechanics......Page 210
4 Some Other Relevant Contributions of Burgatti to Mechanics and Analysis......Page 211
5 Some Conclusions......Page 215
References......Page 216
1 Introduction......Page 218
1.1 The Scuole di Magistero......Page 219
2.1 The Teaching of Projective Geometry in Bologna......Page 224
2.2 Klein's Influence......Page 231
2.3 Epistemological Assumptions at the Basis of Enriques'Vision of Mathematics Teaching......Page 237
2.3.1 A Genetic and Dynamic Vision of the Scientific Process and the Significance of Error......Page 238
2.3.2 Inductive Aspects of Scientific Research and the Dialectic Between Intuition and Rigour......Page 239
2.3.3 Science as a ``Conquest and Activity of the Spirit'' and Unified Vision of Culture......Page 242
2.3.4 The History of Science......Page 244
3.1 The Textbooks for Secondary Schools......Page 247
3.2 The Initiatives of the First Decades of the Twentieth Century......Page 251
3.3 Enriques' Mathesis Presidency and Direction of the Periodico di Matematiche......Page 254
3.4 The Roman Initiatives and Teacher Training......Page 258
References
......Page 277
1 Wave Mechanics and the Schrödinger Equation......Page 284
1.1 ``Derivation'' of the Schrödinger Equation......Page 285
1.2 What About Classical Mechanics?......Page 287
2 The Contribution of Beppo Levi......Page 288
References......Page 295
1 Introduction......Page 296
2 From Bologna to Bologna......Page 297
3 The Transfer to Pisa......Page 302
4 Tonelli in Pisa......Page 305
5 The Relations with Severi and Picone......Page 308
6 Return to Pisa......Page 313
References......Page 321
1 Bruno Pini: A Short Biography......Page 323
2 The Parabolic Harnack Inequality......Page 324
3.1 The Mean Value Property for Caloric Functions......Page 329
3.2 Subcaloric and Supercaloric Functions......Page 330
3.3 The Caloric Wiener-Type Solutions......Page 331
3.4 Caloric Capacity, Heat Equilibrium Potential,Wiener-Type Test......Page 332
4 Final Remarks: The Abstract Parabolic Potential Theory......Page 333
References......Page 337
1 Psychology and Geometry......Page 341
2 The Problems of Science......Page 342
3 The History of Logic......Page 345
4 Logical Controversies......Page 346
5 Conclusion......Page 347
References......Page 348
1 The Complicated Circumstances of the Publication......Page 349
2 Contents and Contributors......Page 356
3 The Contribution of Bolognese Mathematicians......Page 363
References......Page 375
1 Some Biographical Notes of Salvatore Pincherle......Page 379
2 Pincherle and the Mellin-Barnes Integrals......Page 380
3 Pincherle's Foundation of Fractional Derivatives......Page 382
4 Conclusions......Page 385
References......Page 386
References......Page 415
1 Introduction......Page 419
2 Levi-Civita, Pincherle and Amaldi......Page 420
3 Some Notes on Lie's Theory of Differential Equations......Page 424
4 Levi-Civita and Enriques......Page 426
References......Page 429
1 Introduction......Page 431
2 Convolution Equations in Analytically Uniform Spaces......Page 433
3 Regular Functions of a Quaternionic Variable......Page 437
References......Page 441
1 The Context......Page 443
2 Severi's Vorlesungen......Page 444
3 On Riemann's Existence Theorem......Page 446
4 On Moduli of Plane Curves......Page 447
5 On Plane Curves with Nodes and Cusps......Page 449
6 Final Considerations......Page 451
References......Page 452
1 Introduction......Page 455
2 Analytically Uniform Spaces and Surjectivity of Linear Constant Coefficients Operators......Page 456
3 The Case of Real Analytic Functions......Page 458
4 Cattabriga and De Giorgi: Surjectivity Results......Page 462
References......Page 466
1 Introduction......Page 468
2 The Risorgimento Generation......Page 470
3.1 A Biographical Sketch......Page 473
3.2 The New Physics......Page 475
3.3 Beltrami's Reputation......Page 478
4.1 The correspondence Beltrami–Dedekind......Page 480
4.2 Letters by Eugenio Beltrami to Pierre Duhem......Page 484
4.3 A Letter by Beltrami to Wilhelm Killing......Page 489
4.4 Letters by Eugenio Beltrami to Felix Klein......Page 491
4.5 Letters by Beltrami to Rudolph Lipschitz......Page 494
4.6 Letters by Beltrami to Gösta Mittag-Leffler......Page 498
4.7 Some letters concerning Sonya Kowalewskaia's affair of 1886......Page 512
4.8 Beltrami's Report on Sonya Kowalewskaia (1889)......Page 515
References......Page 518
1 Introduction......Page 521
2 Theta functions and their Mellin transform......Page 523
3 Zeta functions machinery......Page 525
4 Quadratic forms......Page 527
5 Lattices......Page 532
6 Integral representation......Page 535
7 Linear systems......Page 537
References......Page 539
Index of Names and Locations......Page 541